First Order Phase Transitions in Lattice and Continuous Systems: Extension of Pirogov-Sinai Theory*

We generalize the notion of "ground states" in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of "restricted ensembles" with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important. An example of a system we can treat by our methods is the g-state Potts model where we prove that for q sufficiently large there exists a temperature at which the system coexists in q + \ phases; ^-ordered phases are small modifications of the q perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all "perfectly disordered" (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the first ̂ -restricted ensembles and of entropy in the last one. Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.